Differential Equations Day 25 -- Picard's Method of Successive Approximations
Overview:
After studying the various methods for solving and numerically estimating solutions to a variety of differential equations, you might wonder if there is any theory that informs the existence and uniqueness of the solutions you have found.
The answer--under certain conditions as described in the following theorem--is "yes". Note: the word "theorem" just means "math fact."
Theorem [Picard, Lindelöf]: For a first order differential equation with initial condition
if and are continuous at , then there exists a unique function solving the differential equation, and matching the initial condition (i.e. ).
The proof of this theorem relies on studying the elements of the so-called Picard Method of Successive Approximations.
The Picard Method constructs a sequence of functions that converge to exactly one solution function of the differential equation with initial condition. The proof is a classic in mathematics because it brings together two different types of mathematical thinking: analysis/topology and algebra. The proof that the sequence converges relies on the Banach Fixed Point Theorem from metric space/topological theory. For the proof of the Picard/Lindelöf theorem about the existence and uniqueness of solutions of differential equations, see here. For the proof of the the Banach Fixed Point Theorem of contractions on complete metrics spaces, see here.
For the remainder of this lesson, we'll focus purely on the Picard Method of successive approximations.
Picard's Method generates a sequence of increasingly accurate algebraic approximations of the specific exact solution of the first order differential equation with initial value. The sequence is called Picard's Sequence of Approximate Solutions, and it can be shown that it converges to exactly one function, , of the independent variable.
In addition to its theoretical importance, Picard's Method also offers an algebraic alternative to numerical methods such as Euler's Method or RK4.
Picard's Method of Successive Approximations:
Given a first order differential equation with initial value
Picard's Sequence of Successive Approximate Solutions is generated by the following formula
where .
The sequence of successive approximations in the case of a specific differential equation can be explored in the applet below.
About the Applet:
The applet below illustrates Picard's Sequence of Successive Approximate Solutions to the differential equation with initial condition
The exact specific solution can be found via the method of separation, and is pictured in purple.
The elements of Picard's Sequence are shown in green.
Slide the variable to see successively more accurate elements of Picard's Sequence converge to the exact specific solution.
We'll work out the algebraic details of the approximations being calculated in the next lesson.
For now, just focus on observing how the green curves (the Picard Sequence of Successive Approximations) become better and better approximations of the purple curve (the specific solution of the differential equation and the initial condition).
On the next page, you'll find a calculator which lets you explore different differential equations and different initial conditions. Check it out!